The MLFFs were then used to calculate the interatomic force constants and
dynamical matrices according to the finite displacement method in a supercell
containing 96 atoms. To account for the long-range dipole-dipole interactions,
the nonanalytic contribution to the dynamical matrix was treated using the
standard method of Ref. [1]}.
As illustrated in Fig. REF , the MLFFs reproduce accurately the AI
phonon dispersion relations for each phase, despite the MLFFs being trained
using a semi-automated on-the-fly procedure, without systematically distorting
the ideal structures around the ground state to reproduce phonon properties.
Some discrepancies in the dispersion of the highest-energy optical phonon modes
can be attributed to finite-size effects. Note that the MLFF trained using BLR
is slightly less accurate than using SVD, as seen in particular for the highest
optical phonon branches in the cubic phase. The RMSE of the phonon energies
reported in Table 1 confirms this trend. It is not surprising
that the largest errors appear in the cubic phase: since it is
unstable at low temperature, the training database contains hardly any
configuration corresponding to this unstable, ideal structure. It is thus harder
for the MLFF to reproduce the 0 K properties of the ideal undistorted cubic phase.
Remarkably, both MLFFs accurately reproduce the imaginary
soft phonon modes in cubic ZrO\(_2\) , which is
essential in order to quantitatively capture phase transitions.
In fact, such modes are generally associated with lattice
instabilities, leading to structural phase transformations in the material.
In ZrO\(_2\) , the soft phonon at the X point in the cubic phase, which involves
a tetragonal distortion of the oxygen sublattice, is linked to the cubic to
tetragonal phase transition [2]}.
